DEVELOPMENT OF COMPREHENSIVE AND INTEGRATED MODELS FOR INERTIAL FUSION CAVITY DYNAMICS

Transcription

1 ARGONNE NATIONAL LABORATORY 97 South Cass Avenue, Argonne Illinois 6439 ANL-ET/-4 DEVELOPMENT OF COMPREHENSIVE AND INTEGRATED MODELS FOR INERTIAL FUSION CAVITY DYNAMICS By A. Hassanein and V. Morozov Energy Technology Division The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory ( Argonne ) under Contract No. W-3-9-ENG-38 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

2 Argonne National Laboratory, with facilities in the states of Illinois and Idaho, is owned by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago. Available electronically at Available for a processing fee to U.S. Department of Energy and its contractors, in paper, from: U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 6 Oak Ridge, TN phone: (865) fax: (865) reports@adonis.osti.gov

3 CONTENTS ABSTRACT.... INTRODUCTION.... TECHNICAL DESCRIPTION BACKGROUND REACTOR DESIGN CONCEPT HEIGHTS-IFE COMPUTER CODE MATERIAL DATA BASE SPACE AND TIME RESOLUTION PHOTON INTERACTION PHOTON SPECTRUM PHOTON DEPOSITION ION INTERACTION MAXWELLIAN OR GAUSSIAN SPECTRA HISTOGRAM SPECTRUM ION DEPOSITION ELECTRONIC ENERGY LOSS NUCLEAR ENERGY LOSS STOPPING POWER TOTAL INTEGRATED ION DEPOSITION FUNCTION LASER LIGHT DEPOSITION THERMAL EVOLUTION OF THE CHAMBER WALL INTEGRATED DEPOSITION MODEL THERMAL RESPONSE MODELS FOR CHAMBER FIRST-WALLS MOVING BOUNDARY PHASE CHANGE EVAPORATION MOVING BOUNDARY EVAPORATION MODELS CONDENSATION MODEL VARIATION OF THERMAL PROPERTIES WITH TEMPERATURE NUMERICAL SIMULATION OF FIRST-WALL TEMPERATURE EROSION PROCESSES PHYSICAL SPUTTERING CHEMICAL SPUTTERING RADIATION-ENHANCED SUBLIMATION EROSION NUMERICAL RESULTS MACROSCOPIC EROSION AND BRITTLE DESTRUCTION CONCLUSIONS... 4 ACKNOWLEDGMENTS... 4 REFERENCES ii

4 FIGURES Figure. Schematic of energy deposition on chamber wall... Figure. Flowsheet for HEIGHTS-IFE package... 6 Figure 3. Photon and blackbody spectra... Figure 4. X-ray deposition, in graphite and tungsten walls in ARIES spectrum... Figure 5. X-ray deposition in composite wall structure from ARIES spectrum... Figure 6. Particle energy distribution for fast ions from NRL direct drive target of 54 MJ yield... 5 Figure 7. Particle energy distribution for debris ions from NRL direct drive target of 54 MJ yield... 6 Figure 8. Tritium target spectrum... 6 Figure 9. General behavior of ion energy loss as a function of ion energy... Figure. Hydrogen electronic stopping power on graphite... Figure. HEIGHTS-IFE calculation of hydrogen stopping power on tungsten and graphite... Figure. Ranges and stopping power for fast and debris ions in cm tungsten wall... 3 Figure 3. Ranges and stopping power for fasts and debris ions in cm graphite wall... 4 Figure 4. Ranges and stopping power for fast and debris ions in mm Li / cm CFC composite wall4 Figure 5. Ion energy deposition for CFC and tungsten... 5 Figure 6. Ion energy deposition, for mm lithium / cm CFC... 5 Figure 7. Properties of CX-U carbon-fiber composite Figure 8. Surface temperature rise due to NRL direct drive target Figure 9. Temperature rise due to laser, X-ray, and ion depositions Figure. HEIGHTS-IFE simulation of RES for hydrogen isotope interaction with graphite... 4 Figure. HEIGHTS-IFE simulation of both chemical sputtering and RES of deuterium on graphite. 4 Figure. HEIGHTS-IFE calculation of physical, chemical, and RES erosion and vaporization... 4 Figure 3. HEIGHTS-IFE calculation of total wall erosion... 4 TABLES Table. Materials in HEIGHTS-IFE package... 8 Table. X-ray deposited yield in Li/Pb and carbon structures... 3 Table 3. Tritium debris histogram... 6 Table 4. Ziegler set of coefficients for lithium, carbon, and tungsten targets... 9 Table 5. ARIES IFE reference target spectra at ns iii

5 Development of Comprehensive and Integrated Models for IFE Cavity Dynamics Abstract by A. Hassanein and V. Morozov The chamber walls in inertial fusion energy (IFE) reactors are exposed to harsh conditions following each target implosion. Key issues of the cyclic IFE operation include intense photon and ion deposition, wall thermal and hydrodynamic evolution, wall erosion and fatigue lifetime, and chamber clearing and evacuation to ensure chamber conditions prior to target implosion. Several methods for wall protection have been proposed in the past, each having its own advantages and disadvantages. These methods include bare walls, gas-filled cavities, and liquid walls/jets. We have developed detailed models for reflected laser light, emitted photon, and target debris deposition and interaction with chamber components and implemented them in the comprehensive HEIGHTS software package. The hydrodynamic response of gas-filled cavities and photon radiation transport of the deposited energy has been calculated using new and advanced numerical techniques. Fragmentation models of liquid jets as a result of the deposited energy have also been developed, and the impact on chamber clearing dynamics has been evaluated. The focus of this study is to critically assess the reliability and the dynamic response of chamber walls in various proposed protection methods in IFE systems. Of particular concern is the effect on wall erosion lifetime of various erosion mechanisms, such as vaporization, chemical and physical sputtering, melt/liquid splashing and explosive erosion, and fragmentation of liquid walls. Mass loss and fragmentation in the form of macroscopic particles can be much larger than mass loss due to surface vaporization and sputtering and have not been properly considered in past studies as part of the overall cavity response and reestablishment. This effect may significantly alter cavity dynamics and power requirements.. Introduction In inertial fusion systems, the power to the first-wall resulting from energetic particles, neutrons, X- rays, and radiation is high enough to cause damage and dynamically affect the ability to reestablish chamber conditions prior to the next pellet implosion. In the case of a dry-wall protection scheme, the resulting target debris products will interact and affect the surface wall materials in different ways. This situation could result in the emission of atomic (vaporization) and macroscopic particles (such as graphite or carbon composites), thereby limiting the lifetime of the wall. Also, mass loss in the form of macroscopic particles can be much larger than mass loss due to the surface vaporization and have not been properly considered in past studies as part of the overall cavity response and re-establishment. This effect could significantly alter cavity dynamics and power requirements. The overall objective of this effort is to create a fully integrated model within the HEIGHTS software package to study chamber dynamic behavior after target implosion. This model includes cavity gas hydrodynamics; the particle/radiation interaction; the effects of various heat sources such as direct particle and debris deposition, gas conduction and convection, and radiation transport; chamber wall response and lifetime; and the cavity clearing. The model emphasizes the relatively long-time phenomena following the target implosion to the chamber clearing in preparation for the next target injection. It takes into account both micro- and macroscopic particles (mechanisms of generation, dynamics, vaporization, condensation, and deposition due to various heat sources: direct laser/particle

6 beam, debris and target conduction, convection, and radiation). These processes are detrimental, and their minimization is of significant importance to the success of IFE reactors. The developed models for HEIGHTS-IFE are modular and can easily be upgraded to multidimensional. The main model consists of a core including the input/output interfaces, the geometry definition, and the numerical solution control. Several modules define the physics of the different phenomena involved. These modules allow radial energy deposition calculation of X-ray and particle debris behavior, including wall deposition and transient heat transfer such as melting and vaporization/boiling of first-wall materials. The current model focuses primarily on the dry wall concept with or without gas protection but is designed to enable future design for the liquid film and/or liquid jet protection schemes, as may be required. The experience gained from earlier use of the HEIGHTS-MFE package, which contains unique models and physics for magnetic fusion energy, was directly applied to simulate the dynamics of chamber behavior in inertial fusion reactors. Various aspects of the HEIGHTS-MFE models were benchmarked and tested against worldwide simulation devices and tokamak reactors in Japan, Europe, Russia, and the US. Besides magnetic fusion research, the HEIGHTS package has been used and is currently being applied to the space program (fire & ice projec, high-energy physics program (muon collider and neutrino factory projects), nuclear physics program (ATLAS projec, and medical (isotope production and arc injury) and possible defense applications (Missile Defense System). The present study has resulted in a fully integrated, efficient model directly created for and applicable to IFE chambers. The HEIGHTS-IFE code can be used to simulate chamber dynamics in studies such as ARIES-IFE and can also help plan and understand pre-ire or IRE experimental results regarding IFE chamber behavior.. Technical Description Following the micro-explosion in an IFE reactor, high-energy X-rays and ions are produced and travel toward the chamber wall at high but differential speeds. Some of their energy is deposited in the residual or protective chamber gas and is re-radiated to the wall over a longer time. Figure shows a schematic of the energy deposition on the chamber wall. The main HEIGHTS code models the transient hydrodynamics of the chamber gas, focusing on the relatively long-time phenomena following the target implosion to the chamber clearing in preparation for the next target injection. The resulting energy deposition over time on the chamber wall can be used as input in the wall-interaction computer module. VAPOR First Wall Incident Beam Target Fragments Depositions Structure X-rays Neutrons Ion Debris Laser Figure. Schematic of energy deposition on chamber wall

7 Specific of the processes that are modeled include:. Energy Deposition As a result of thermonuclear burn in inertially confined fusion (ICF) reactors, the first-wall could be exposed to photon radiation and ion beams with a wide range of energies. The energy deposited in a material can be calculated from the mathematical relation for energy loss for each radiation type. This study contains comprehensive analysis of these processes, including energy deposition from photons, ions, and laser beams.. Thermal Evolution The thermal response of the chamber wall exposed to thermonuclear radiation is determined when the time-and space-dependent energy depositions are known. 3. Melting/Phase Change Melting can occur in the case of a solid metallic during intense deposition of energy. Complexities in modeling this process arise as to the behavior of the melt layer under different loads and the resulting wall material loss. Mechanisms that contribute to melt layer loss are partially known and include effects such as melt splashing due to formation, boiling of volume bubbles that may result from continuous heating, and overheating of the liquid and other possible instabilities driven by various forces such as shock loading and gravity []. Laboratory experiments on the effects of high heat fluxes and beam deposition on target materials have shown the formation of numerous liquid droplets that are splashed and lost during beam-target interaction. 4. Evaporation/Sublimation Evaporation/sublimation increases substantially once the vaporization or sublimation temperature is reached. However, material vaporization also occurs at lower temperature and is dependent on the local conditions, such as surface temperature and the partial vapor pressure. The melt layer thickness is usually much larger than the surface vaporization. Therefore, splashing erosion from the developed melt layer could be quite important in determining the lifetime of IFE chamber walls. 5. Physical Sputtering Particle sputtering can be important in certain situations depending on the impacting ion energy and chamber conditions. A physical sputtering module has been developed to calculate chamber wall erosion due to particle debris bombardment. Physical sputtering can be an important erosion mechanism in ICF reactors. 6. Chemical Sputtering Formation of volatile molecules on the target surface due to a chemical reaction between the incident particles and the target atoms is called chemical sputtering. It is especially observed for hydrogen and oxygen bombardment of graphite and carbon-based materials by the formation of hydrocarbon molecules, such as CH 4 and CO. In contrast to physical sputtering, chemical erosion strongly depends on the target temperature. 7. Radiation-Enhanced Sublimation In carbon and carbon-based materials (CBMs), besides the enhanced temperaturedependent erosion yields by chemical sputtering around 8 K, enhanced erosion yields were measured during ion bombardment at target temperatures above K. This effect 3

8 is known as radiation-enhanced sublimation. A model has been developed and implemented to calculate this effect as a function of wall temperature. 8. Macroscopic Erosion Macroscopic erosion may result from melt layer splashing of metallic components as well as from spallation (solid fragments) of brittle and carbon-based materials. In laboratory simulation experiments to study the effects of plasma instabilities in tokamak devices, macroscopic erosion dominated the erosion loss mechanism. Melt layer splashing occurred due to both melt layer superheating and hydrodynamic instabilities. Both mechanisms can exist during inertial fusion cavity response. Although macroscopic erosion is very complicated to model because many processes are involved, we intend to develop a module that build on the experience gained from the earlier magnetic fusion work. 9. Condensation and Redeposition These processes must be taken into consideration to provide an accurate net mass loss at each time step as input for the main hydrodynamic code. The actual condensation and redeposition rate of wall material will depend on the cavity conditions, as well as the type of erosion products. The interaction and redeposition of macroscopic erosion products are complicated, and initial models are being developed to assess the geometrical effects of the cavity chamber on overall net wall erosion and on cavity clearance before the next microexplosion reaction. The present effort started by assessing the importance of the above processes under various conditions. For example, sputtering might not be important over a certain energy threshold and cavity conditions. This assessment allowed us to focus our code development efforts on the key processes to start with, then mathematical models and equations were evolved to simulate the essential physics. The progress made in modeling these processes is present in sections. The proposed tasks and deliverables for FY were as follows: o o o o o o Identify all potentially important processes for IFE chamber wall interaction. These processes include: Energy deposition from photons, ions, and neutrons Spatial heat generation during various deposition phases Melting Evaporation/sublimation Physical sputtering by target debris Chemical sputtering Radiation-enhanced sublimation Macroscopic erosion due to spallation Assess the importance of these processes for IFE chamber wall interaction by performing preliminary parametric estimates. Identify key processes for inclusion in the models. Evolve mathematical models and equations capturing the essential physics of these key processes. Write an independent, integrated model based on above models. Perform preliminary comparison of calculated results with existing codes for selected cases. 4

9 o Plan for future upgrade of computer code to add more detailed models and cover a wider range of conditions for use in design studies of the IFE reactor chamber. 3. Background In the last decade, Argonne has developed substantial experience in modeling beam target interaction under intense energy deposition. In magnetic fusion, for example, plasma-facing components (PFCs) have to accommodate peak transient energy deposition resulting from off-normal plasma conditions. Short-term plasma disruptions depositing energy (~- MJ/m ) on PFCs over ~.- ms result in loss of wall and divertor material through vaporization and melting but with little effect on the heat flux through the structural substrate interface to the coolant, whose time constant is much higher. In addition, longer-term energy deposition following other plasma instability events can result in plasma energy deposition of ~- MJ/m over.- s duration. These slower transient events can also result in substantial component vaporization and melting. In addition, since the energy deposition time is of the order of the component's thermal diffusion time constant, high heat fluxes could penetrate through the wall/heat sink interface to the coolant. Different computer models have been developed to help understand the effect of these high transient energy depositions on the wall components. These include more fine-detail physics implemented in comprehensive and integrated models such as the HEIGHTS package developed at ANL [3-9], the structure of which is schematically presented in Figure. Many of the processes involved are similar to the case of the IFE chamber, and the experience in developing the HEIGHTS package was a major asset in evolving this IFE-chamber-specific wall interaction code. 4. Reactor Design Concept Currently proposed concepts for inertial confinement fusion reactors have several design and operation features. Each concept employs a blast chamber in which the thermonuclear microexplosion occurs and is contained. Laser light or ion beams provide the heating and compression of the fuel pellet to ignition temperatures and are directed into the blast chamber from final mirrors or focusing elements through ports located on the periphery of the cavity. As a result of the reaction, various fusion products are emitted and could impinge upon the blast chamber wall if the chamber is pumped to a hard vacuum. The thermonuclear burn of the fuel and the subsequent emission of fusion products, which strike the first-wall, occur over a very short time (less than ns). As a result, a large amount of energy is deposited in the wall in very short time and hydrodynamic stress waves are produced. One effect of the rapidly repeated microexplosions is to quickly deteriorate any unprotected solid surfaces of the blast chamber. Therefore, some type of first-wall protection may be needed to maintain the structural integrity of the blast chamber. At the same time, the main objective of the ICF reactor is to efficiently convert heat, which is generated in the blast chamber and surrounding blanket, into usable energy. In addition to shielding the blast-chamber first-wall, the protection system must permit rapid recovery of the energy in a form that is suitable for utilization in the energy conversion cycle. Thus, the first-wall protection system establishes many of the reactor design characteristics. Most current ICFR designs assume that the fuel pellet will contain a deuterium (D) and tritium (T) mixture, as well as some low Z ablator (e.g., C, O) and high Z (e.g., Fe, Ta, Pb) elements. The fuel is compressed to the required conditions of temperature and density by the beam. The surface of the target is violently heated and ablated by intense beams. Very high pressure is generated, accelerating the fuel inward. The high Z material carries kinetic energy away from the microexplosion and moderates the alpha particles emitted as a product of the reaction. Ignition occurs when rapidly 5

10 moving the pressure generated in the compressed matter suddenly stops at the inner region of the fuel and ignition temperatures are reached. Ion Debris X-Rays Laser Beams Spectrum Calculation Maxwellian Gaussi an Spectrum Calculation Blackbody Histogram Histogram Analytical Models Semi-empi rical Models Cross-Sections Library Li ght Ions Heavy Ions Deposition Heat Flux Ion Displacement Thermal Res ponse Moving Boun daries Variable Properties RadiationLosses Sputtering Finit e Difference Finite Element Convective Cooling Ch emical Sputtering Kinetic Multispecies Radiation Transport Model Ra diation- Enhanced Su blimatio n SPLASH Hydrodyn amic Melt- LayerStability Redeposition Calculations Phase Change Evaporation/ Condensa tion Net Erosion Gas/Vapor Shielding MultigroupRadiation Transport -D Capabilities Forward-Reverse/Ray Method Con tinuum (4 Groups) Line ( Groups/Line) Ful l Gas Hydrodynamics Dynamic Stress Cavity Response Lifetime Analysis Figure. Flowsheet for HEIGHTS-IFE package For a simple, bare fuel pellet microexplosion, the energy released is partitioned among different species: X-rays, reflected laser light, alpha particles that have escaped the plasma, plasma debris, and neutrons. The plasma debris consists of both fast and debris ion fluxes. The energy of this debris in 6

11 nearly a Maxwellian distribution with an average energy equal to the energy deposited in the pellet from the laser or ion beam plus the fraction of the thermonuclear reaction energy divided by the number of pellet particles. Also in case of a laser-driven system, the laser light will contribute to the total energy released through a reflection mechanism. The energy spectrum for the X-rays may vary over a wide range. Thus, the energy partition and energy spectrum depend upon the pellet design. Energy deposition by X-ray, fast and debris particles occurs on, or very near, free surfaces of incidence in structural and coolant materials, whereas the energy of neutrons is deposited throughout relatively large material volumes. The interior surface of the cavity wall would have to withstand repeated high yield energy deposition, and a very high surface temperature increase would result. Tolerable surfacetemperature increases of such structural components have not been established either theoretically or experimentally. 5. HEIGHTS-IFE Computer Code The structural programming procedures of HEIGHTS-IFE are designed to analyze the temperature increase and erosion of the cavity wall from pulsed thermonuclear radiation. The package was developed to study energy deposition, thermal evolution and temperature response, evaporation, sputtering, and other subsequent effects produced in materials by transient pulses of photons and ions. The models within HEIGHTS-IFE are sufficiently accurate and efficient to allow simultaneous analysis of a wide range of ion and photon spectra, which may be arbitrary specified. This report contains an overview and description of the major features of the HEIGHTS-IFE package. Package use is demonstrated with target spectra from the Naval Research Laboratory (NRL) and data derived in the ARIES Program. The ARIES Program is a national, multi-institutional research activity dedicated to advanced integrated design studies of long-term fusion energy, with the ultimate goal of identifying key R&D directions and developing advanced systems for the fusion program. 5.. Material Data Base We are attempting to solve the problems of ICF engineering design and the choice of construction wall materials which are able to withstand the influence of high-energy radiation flux, temperature rise, and erosion rate within the thermonuclear reactor environment. Different materials have been studied, but all of them have their strong and weak points. A comprehensive database for potential candidate materials has been assembled and incorporated in the HEIGHTS-IFE package. In the computer simulation, each material is described by its physical and chemical properties. We keep our set of properties as self-contained as possible. Our collection of properties is tabulated and fitted for materials which are of major interest. Data for properties are best fitted over a wide range of temperatures and conditions relevant to ICF reactor design. The database structure for new materials is quite easy to implement. Each material has its own unique name, by which it may be chosen as a wall candidate. Currently implemented materials and their names are presented in Table. 5.. Space and Time Resolution The major goal of this analysis is to evaluate the near surface evolution of the chamber wall, which must take into account the temperature rise, erosion rate, physical and chemical sputtering, radiationenhanced sublimation, evaporation, and melting. Temperature wall distribution is computed as a function in space mesh points in the computer code. Therefore, the spatial mesh distribution must be 7

12 Table. Materials in HEIGHTS-IFE package Long Name Short Name Code Case stainless steel ss molybdenum mo beryllium be 3 carbon (common) c 4 irradiated graphite c h45i 4 nuclear grade graphite c h45 4 graphnol c n3m 4 pyrolitic graphite c pyr 4 3 pyrolitic compressed graphite c pyr c 4 4 CFC A5 C/C (carbon fiber composite) c cfc a 4 5 CFC CX-U / jap c cfc ll 4 6 CFC CX-U / ll c cfc ll 4 6 CFC CX-U / pr c cfc pr 4 7 CFC SEPCARB N c cfc n 4 8 CFC SEPCARB / IRR c cfc ni 4 8 lithium li 5 gallium ga 6 tin (Sn) sn 7 LiSn lt 5 water wt 9 beryllium oxide bo SiC sc aluminum al copper cu 3 Dispersion-strengthened Cu cu-al5 3 tungsten w 4 W ITER w iter 4 vanadium v 5 tantalum ta 6 niobium nb 7 lead pb 8 8

13 treated very carefully. A uniform mesh is sufficiently accurate for most applications, providing stable and fast calculation. At the same time, our physical problem requires that particular attention should be paid to calculation near the surface, particularly when the composite wall changes its properties. It is also possible to use a combination of uniform and logarithmic nonuniform space mesh. A user should choose the number of zones in the mesh and the type of each zone. The mesh is generated once and then is used for all further calculations. 6. Photon Interaction The first-wall of ICF reactors can encounter photon radiation levels which range from a few ev to a few MeV. The primary interaction of photons with materials in these energy ranges include: photoelectric effect coherent scattering incoherent scattering pair production Cross sections for each of these interactions have been tabulated in various forms and are available for numerical calculations. At low photon energies the total photon cross section is dominated by the photoelectric cross section in which a photon transfers all its energy to an electron in the vicinity of a nucleus. The electron is emitted (Auger electron) with the photon energy minus electron binding energy. The cross section for this interaction shows a very strong material and spectrum dependence. A simple and convenient equation for the photoelectric effect has been proposed by Biggs and Lighthill []: 4 k σ j = C jke cm k = g, where a set of parameters, C jk, is used for fitting the data within discrete energy intervals characterized by the parameter j. The spectrum is broken into different intervals to properly account for absorption edges. At the high-energy end of the spectrum, pair production will be the dominant contributor to the total cross section. The pair production process is a photon-matter reaction, which occurs when the electric field of the photon interacts with the electric field of an atomic nucleus. The incident photon is destroyed, and a positron-negatron pair is created. The reaction occurs when the incident photon energy is higher than or equal to threshold energy of mc (. MeV). The interaction rate depends on the nuclear cross section and is, therefore, proportional to Z of the absorbing material. The differential cross section in relation to the energy shared by the positron and the total cross section; obtained by integrating over all positron energies, have analytical expressions and accurate approximations. Since the process is a nuclear interaction, the cross section is simply proportional to the nuclei density and Z. At intermediate energies the principal photon interaction can be incoherent (or Compton) scattering. In this process, energy is given by an incident photon to an electron and results in a scattered photon. The incoherent scattering cross section can be derived using quantum electrodynamics and is given by the Klein-Nishina formula for unpolarized incident radiation as: [ ( )] 3 + cos ϑ + χ ( cosϑ ) + χ( cosϑ ), where dσ dω = r + χ cosϑ 9

14 dσ cm differential cross section, dω electron E χ = ; m c r classical electron radius, ϑ scattering angle; E photon incident energy This equation represents the cross section for one electron, and since Compton scattering implies incoherent field superposition, each electron adds independently. Thus, for a given material, the above formula is multiplied by atomic number to obtain the differential cross section per atom. Integration can give the total cross section, which is difficult to utilize for numerical evaluation. However, Biggs and Lighthill [] give a useful approximation as: σ 3 cm; Z +.48χ +.64χ cm.46 A + 3.7χ +.938χ +.575χ g. = 3 Coherent scattering occurs when the energy of the incident photon is reduced to low enough frequencies where the momentum can be ignored. The classical formula for Thompson scattering for d σ r isolated electrons gives = ( + cos ϑ). Integrating over all directions, the total Thompson dω a 8 scattering cross section is obtained: = π barns σ r =. 665 in. Since coherent scattering is 3 electron elastic, it does not result in any net loss of photon energy, and no significant local deposition of energy occurs. The package calculates the volumetric energy deposition for X-ray spectrum or monoenergetic photons. The spectrum may be specified as blackbodies or in histogram form. Deposition is based on general photoelectric and incoherent cross-section libraries, which have been incorporated into the project. For high-energy photons, the photoelectric cross-sections are negligible compared to the cross sections from incoherent scattering. The total incoherent cross sections are used in this case. Temperature calculations are done for the adiabatic case, an impulse solution, and a finite duration deposition. 6.. Photon Spectrum The photon response of a first-wall can be determined if the photon spectrum is specified. However, this spectrum depends on the target design and can only be described by very sophisticated and, therefore, quite expensive methods of calculation. The response of the wall can also be determined if the photons are characterized by common spectral forms. The wall loading can then be found by determining the spectral dependence of the energy deposition. A commonly used spectrum for low energy photons is the blackbody of the Planckian spectrum, which is used when radiation emission is specified by the temperature of the emitter. The mathematical representation of the blackbody spectrum is given by

15 5F S( E) = kt π E U = ; kt 3 U e 4 U kt - characteristic energy, kev; J cm kev, where F - total fluence or energy density, J cm. The wall loading from source photons will occur at a time equal to the cavity radius divided by the speed of light. This is only true for a medium where the dielectric constant is independent of the frequency, so that the propagation of all energies will be at the same velocity. The deposition time for the X-ray energy spectrum is assumed to be between and ns. The deposition of X-rays into first-wall materials will strongly depend on the energy spectrum of these X-rays. Soft X-rays deposit their energy within a micrometer of the wall's surface, very rapidly heating a thin layer of the first-wall to a higher temperature. Harder X-ray energy spectra penetrate relatively larger distances into the material, therefore heating a larger mass to a lower temperature. NRL target out represents photon spectrum information in the form of the three temperature fitting functions: E(T ) = c T 3 T T e + c T 3 T T e + c 3 T 3 T T e 3 J kev, where T photon energy in kev and parameters c, T, i =, K, are i i 3 c T c T c3 T3 5 3 NRL Direct HI Indirect Figure 3 compares photon spectra from direct and indirect target output, as well as a kev blackbody (BB) spectrum Figure 3. Photon and blackbody spectra

16 6.. Photon Deposition In our numerical simulation, the deposition function and the total X-ray yield that is deposited in the wall were calculated by two methods using different space meshes. Three types of meshes may be generated: standard space mesh that can also be used for ions, new linear type mesh with increased/decreased number of points, and new logarithmic space mesh with changed number of points. The goal is to check the influence of each layer on the amount of X-ray absorption. To compute wall thermal evolution and its temperature rise further, standard space mesh results are used. Numerical simulation results of target implosion were obtained by means of ARIES spectra information for an NRL direct drive target (total yield.4 MJ) deposited in both a carbon fiber composite (CFC) and tungsten wall. As shown in Figure 4, the CFC material allows X-rays to penetrate more deeply through. As a result, a lower the temperature rise is expected. The HEIGHTS-IFE package can represent the chamber wall in a multicomponent structure. For example, Figure 5 and Table show the absorption of NRL photon spectra in composite Li/Pb film and carbon structure. Figure 4. X-ray deposition, in graphite and tungsten walls in ARIES spectrum Figure 5. X-ray deposition in composite wall structure from ARIES spectrum

17 Table. X-ray deposited yield in Li/Pb and carbon structures Li C Absorbed Total NRL Direct.9 MJ.8 MJ.74 MJ.4 MJ HI Indirect 79 MJ 3 MJ MJ 5 MJ Pb C Absorbed Total NRL Direct.35 MJ.5 MJ.4 MJ.4 MJ HI Indirect 4.73 MJ. MJ 4.74 MJ 5 MJ 7. Ion Interaction 7.. Maxwellian or Gaussian Spectra Ion deposition was calculated by means of several comprehensive models to predict the behavior and the slowing down of the incident ion flux in various candidate wall materials. In addition, sufficiently reasonable approximations were made by using Maxwellian or Gaussian distributions as well as a histogram. Tabulated results were easily accommodated in HEIGHTS-IFE in the form of histogram input to these calculations. A Maxwellian distribution is characterized by a mean energy E m in the form: S ( E) = E m N E E π m e E E m, kev where: E m characteristic energy in kev; E ion energy in kev; N total number of ions in ions /cm. When a spectrum of a specific width is required, the Gaussian distribution is used. The mean energy E m and the standard deviation σ are necessary to describe the distribution: SE ()= N πσ e E E m σ where: E m characteristic energy in kev; E ion energy in kev; kev, N total number of ions in ions /cm ; σ standard deviation in kev. When the energy distribution is known, the number of points N and total amount of incoming ions energy Y, can be calculated from the following: N = S( E) de in and Y = in kev. cm S ( E) EdE The uniform distribution of energy E in kev is calculated in a given range [ E min, E max ]. E E 3

18 7.. Histogram Spectrum For some studies, target computer simulation predicts output spectra in the form of tabulated outcome. The NRL target simulation code supplies ion spectrum information in the form of a histogram. The histogram spectrum shows the distribution of incoming flux depending on the energy of particles. Typical output ion spectra from target implosion calculations are shown in Figure 6 and Figure 7. These spectra are the result of NRL simulation of a thermonuclear target implosion with a total yield of 54 MJ. Table 3 and Figure 8 present an example for tritium ions. The total amount of incoming energy (yield) for these tritium ions is. MJ. Substituting tritium beam parameters and integrating time flux by arrival time t T gives the total number of incident particles N = Similarly, the power, converted from kev to MJ, gives the total yield of the beam equal to 3.74 MJ Ion Deposition When the ion flux arrives at the wall, it starts to penetrate through the condensed wall material and loses its energy as it slows down. The theory of the energy loss of particles in matter dates back to about 93. Since that time, the study of particle stopping power has been an active area of theoretical and experimental research. The interaction of light ion beam (H, D, T, He 3, He 4 ) with the wall has been carefully studied by many authors because of its significance for fission and high-energy physics applications. The approximate behavior of energy loss for ion beams consisting of heavy ions is sufficiently understood. Nonetheless, in numerical simulations, quantitative data may vary significantly for different methods. Below, we describe the most often used ways to model energy loss in computer simulation: A collection of experimental data sets for a particularly chosen pair of ion beam-wall material is approximated by an analytical expression. This method has very high accuracy for the slowing down approximation only if the required range of initial energies is overlapped by some points in the experimental data sets. The efficiency and the accuracy of this approach are quite good. Unfortunately, the amount of experimental data is limited to incident H-, and He- ion beams and inappropriate for the others. Theoretical investigations allow one to predict the behavior of energy loss from incident energy in bound cases. The well-known Bethe- Block formula [3], [4] gives a sufficiently accurate approximation of energy loss for a higher incident energy beam, while the Lindhard work [5] provides a sufficiently accurate approximation for the lower energy ions. The major problem with the theoretical modeling is the transition cases, when neither of these models works appropriately well. The problem of accurately calculating spatial details of the stopping power (i.e., how much energy is lost while an incident ion is moving through the wall) is still not well resolved, despite many authors suggesting numerous methods based on theoretical works, experimental results, or a combination of both. The two main concerns are insufficient experimental data for low-energy incident 4

19 particles and the diversity of possible variations of the incoming fluxwall material. These concerns have led to numerical approximations of the energy deposition function based on experimental data extrapolated to unknown combinations of incoming ions-wall material. At the same time, experimental data often covers a broad range that increases the level of uncertainty so that fitting methods are employed to smooth the data. Figure 6. Particle energy distribution for fast ions from NRL direct drive target of 54 MJ yield 5

20 Figure 7. Particle energy distribution for debris ions from NRL direct drive target of 54 MJ yield Table 3. Tritium debris histogram Energy, kev Ions/keV E E E E E E E E E E E E E E E+6 Figure 8. Tritium target spectrum For most heavy-ion cases, there is insufficient experimental data for accurate fitting. At the same time, experiment data allows us to scale theoretical formulas and predict the energy loss for a whole range of energies. The accuracy of this method may be from 5-%. Having a reliable description of the deposition behavior of one ion beam with the wall gives a way for scaling its behavior to the 6

21 interaction of another ion beam with the same wall material. Based on proton-solid, proton-proton, and proton-gas cases (the most carefully studied cases with sufficient amount of reliable experimental data), the same behavior scaled by some factor is assumed for other incident ions. Usually, light ion interaction is implemented separately, while heavy ion cases are simulated by means of scaling to the proton interaction function. The interaction of charged particles with materials primarily involves two processes. The first interaction is between the incident ion and the electrons in the wall material, which is an inelastic collision. The second interaction is between the collisions of the ions with material nuclei, which are an elastic interaction. The dominant mechanism of ions slowing down in materials depends upon the instantaneous energy of the moving ion Electronic Energy Loss The slowing down of a charged particle due to interaction with electrons in a material is usually divided into three energy regimes, i.e., high, intermediate, and low. In the high energy regime, the particle velocity exceeds the velocities of the orbital electrons. In the intermediate energy regime, these velocities are of the same order. In the low energy regime, the particle velocity is much smaller than the orbital velocities of the material electrons. To date, all simulations of ion beam energy deposition that have included temperature effects have used the Bethe equation to describe the bound electron stopping power. This formula accounts for both ionization and excitation of the atomic electrons and is widely used for the high-energy region. The formula is usually written in the form: de dx Bethe z z 4πN zeff ρe = m c β a e 4 z mec β γ ln β I atomic number of the projectile ion; atomic number of the stopping matter; Ci, where z z eff effective charge of the projectile ion, approximated by z ; N 3 Avogadro number, mol atomic mass of stopping matter, amu; a m e c 5 electron rest energy, ev; e electronic charge squared, ev Å; g ρ density of the stopping matter, 3 ; cm β C i i z i nondimensional ratio of projectile ion velocity to velocity of light; sum of the effects of shell corrections on the stopping power; - ; β γ. 7

22 The average ionization potential used in the Bethe equation is formally defined by z ln I = n f n ln En, where E n and f n are possible electronic energy transitions and corresponding dipole oscillator strengths for the stopping medium. Term I is the function of the target only and independent on the incident ion velocity. Ziegler [8] have evaluated this expression and tabulated it for various solid-state charge densities. Note that if one ignores the polarization effect correction term, which is important, only when energy is extremely high (above GeV/amu) and shell correction term, then the logarithm approaches zero and the stopping power becomes singular when m e c β γ I. Equivalently, the lower bound of the applicability of the uncorrected Bethe equation for the projectile energy is I E > 9389 a + m e c, where energy E is given in kev. For example, lithium has a tabulated average ionization potential of 47.6 ev. Therefore, interaction of the projectile ion with the lithium 5 wall by Bethe theory is limited for the velocities β > or, substituting tritium parameters, E > 65.4 kev. Analogously, the limit of Bethe theory for protons is E >.86 kev. Even with the inclusion of shell corrections, the Bethe theory is not appropriate for very low energy ions. In this regime, Lindhard and his coworkers [9] have developed the LSS model in which the particle energy loss is proportional to its velocity and is usually presented as: d ε = k ε, where dρ E ε = non-dimensional reduced energy; E L R ρ = non-dimensional reduced length; R L k =.793z /3 z e z e z ( + A) 3/ ; z /3 /3 ( + z ) 3/4 a a a projectile particle charge; target charge; A = ratio of target mass and projectile particle mass; + A zze = in ergs; A a E L ( A) + R L = in cm; 4 π ANa.4683 a = z + z / 3 8 / 3 in cm; 8

23 atoms N target atom density in. 3 Å The LSS model is valid when the particle velocities are below the orbital velocity of the target 3 electrons. This gives the upper bound of its applicability as E < 4.97 z a kev. The intermediate energy regime between the upper limit of the LSS model and the lower limit of the Bethe-Block equation has no basic theoretical treatment at the present time. According to several authors, this region may be described by transitory functions based on LSS and Bethe-Block. Varelas and Biersack [3] suggested the formula = +, where S LSS is the stopping power from the S S LSS S BB LSS model, and S BB is that from the Bethe-Block equation. 4 For computer simulation, simple fitting curves, based on both theoretical descriptions and experimental results are more convenient. Ziegler [7] has compiled data for the electronic stopping of helium, and Andersen and Ziegler [6], for hydrogen in all elements. For example, the hydrogen electronic stopping power is given as ev S H = A E, 5 atoms cm S H = +, S S LOW HIGH.45 A3 A4 S LOW = A E, S HIGH = ln + + A 5 E, E E S H 4 A A 7β = 6 ln β A i β β i= + 8 i ( ln E), ev for E [, ] kev [ ] for E, 999 kev 5 atoms cm ev 3 5 for [, ] 5 atoms cm E kev kev where E - the ratio of hydrogen energy to hydrogen mass in, β - the ratio of projectile ion amu velocity to velocity of light, and A K A - set of tabulated data values, dependent on the target element, presented in Table 4 for lithium, carbon, and tungsten targets. Table 4. Ziegler set of coefficients for lithium, carbon, and tungsten targets A A A 3 A 4 A 5 A 6 A 7 A 8 A 9 A A A Li.4E.6E 7.56E 3.3E E-.53E-3.47E E- 5.6E- -.83E- 9.98E E-4 C.63E.989E.445E3 9.57E.89E- 3.59E-3.3E E.44E E-.E E-4 W 4.574E 5.44E.593E E 3.44E E-3.475E E 5.39E E- 3.48E E-4 9

24 As mentioned above, incident light ions, with energies higher than a few kev, lose their kinetic energy in materials mainly due to electronic interaction. This observation is true for light ions that are present in ICF reactors such as helium, deuterium, and tritium. The implementation of the following model is based on the work of Hunter [3]. He suggested that the energy loss may be accurately described by means of a similar parametric set of expressions. Parameters may be set up by using four data fit points for each particle pair of incident ion wall combination. He also developed a set of analytical formulas for the spatial distribution, which could be evaluating from the electronic energy loss data of Brice [3]. The stopping power data were divided into three regions similar to the energy regions shown in Figure 9. Each region has a function that reproduces the data well. These functions are given by: de dx de dx de dx ( E) = S E E β E ( E) α ( e ) Region, < E E, = Region, E < E E, E δ ( E) γ e = Region 3, E < E, where S,E, α, β, γ, and δ are all constants. The constants for each region can be determined by selecting reference points ( E, S ), ( E, S ), ( E, S ), and ( E ) 3, S 3 from the stopping power curve, which in turn could be determined from the Brice formulation. de dx S E,S S S 3 I II III E E E E 3 E Energy, kev Figure 9. General behavior of ion energy loss as a function of ion energy To achieve a smooth transmission from the region I to region II regime, one determines S log S α =, α β =. S S E

25 Analogously, to construct a smooth transmission from the region II to region III regime, one sets Finally, one needs to find the point E E3 E δ δ =, γ S e log( S / S3 ) (, S an iterative method of solving the nonlinear equation iteration. Here, =. of intersection for region I to region II, this is done by E ) j j+ j f ( x ) x = x f ( x j ) δ e x x β x β γ x δ ) α α e γ f x) αβ e e f ( x =, ( = +, x = E. δ, known as the Newton We have implemented all described methods as separate routines to benchmark each method against the other. Figure, shows the hydrogen stopping power in the graphite wall predicted by several theoretical models. The left graph combines the set of analytical approaches. The low-energy region is represented by the LSS model which is valid for energies less than 5 kev. The high-energy region is represented by the Bethe model, which is valid for energies more than 35 kev. The gap between these models is shown as a three-point spline approximation. The right graph compares two models: semiempirical Hunter set of approximations and Ziegler fitting function based on experimental results. HEIGHTS-IFE uses the combination of all described methods. Figure plots the hydrogen stopping power in both tungsten and graphite calculated with HEIGHTS-IFE. The curves indicate that both models predict analogous values for stopping power for a sufficiently wide range of computational parameters. Figure. Hydrogen electronic stopping power on graphite

26 Hydrogen on Tungsten Hydrogen on Graphite Figure. HEIGHTS-IFE calculation of hydrogen stopping power on tungsten and graphite 7.5. Nuclear Energy Loss The second mechanism of slowing down a charged particle is the elastic collision of these particles with material nuclei. The rate of interaction will be determined through the nuclear cross sections. A relatively simple analytic expression for the nuclear cross section was derived by Lindhard and Scharff [5] using a shielded Coulomb interaction with a Thomas-Fermi atomic model. Stewart and Wallace [33] have performed a least-square fit to the Lindhard approximation. They have found that the nuclear energy loss can be expressed as de.77 ' 45. C e ( Cnε = ) n ε dr nuc MeV g cm, where R = ρ x, ρ density of the media in g cm 3,.75 C 6 A A 3 3 ZZ = 4.4 Z Z n A A ZZ + + A +.5 ' = A A 3 3 ( ) + + Z E MeV C n Z A A Z, ε =. Z A amu ( A A ) Here, Z, A are projectile atomic number and atomic mass, Z, A are media atomic number and atomic mass Stopping Power Below are the results obtained from HEIGHTS-IFE numerical calculations of the stopping power for the NRL target spectra (referenced in ARIES website) for wall materials of carbon and tungsten. Figure illustrates the loss of fast and debris energy in the tungsten wall, while Figure 3 shows the same but for the graphite wall. HEIGHTS-IFE is also able to simulate composite wall structures. Figure 4, for example, presents the numerical simulation for a lithium/cfc composite wall bombarded by a high-energy yield of 9.5,

27 MJ (fast ions) and a low-energy yield of 4.9 MJ (debris ions). Ion spectra were also taken from the NRL direct drive target Total Integrated Ion Deposition Function Particular attention must be taken when computing the total ion deposition function since, for each separate ion beam, the deposition is computed in local ion time, which differs for each particular ion. Nevertheless, the total ion deposition function should combine the individual contribution of each ion beam correspondent to the arrival and termination time. This means that all individual local ion times should be converted to a standard time, after which each ion deposition function is converted from ion to standard time. To simplify the conversion and to preserve the accuracy of the calculated deposition function, the standard time is generated from local ion time arrays. In such a way, the standard time mesh is an ordered array, combined with the points of local ion time meshes, so that starting and ending points for each ion belong to the standard mesh array. To begin, starting and ending time points are found for each ion. These points must be present in the standard time mesh because they bound arrival time for some particular ion. Next, all points for all time meshes are changed to standard time mesh. Particular attention is paid to exclude starting and ending points if they occur twice. Finally, if the number of points becomes too large, then some points are eliminated. The last step reorders the standard time mesh from the lower to the largest time points in the mesh array. When standard mesh is generated, it is straightforward to find the time periods of each ion s presence and absence within the array. Let us assume that ion arrives from time to time, and ion arrives from time t to time t. Then, one may have either a single consistent range or two inconsistent time ranges of ions coming, depending on the t values of the ion ranges. The set of possible situations is shown below. The first and the last cases correspond to two instantaneous ranges, while the rest of the cases are combined as one consistent range. t t Fast Ions Debris Ions Figure. Ranges and stopping power for fast and debris ions in cm tungsten wall 3

28 Fast Ions Debris Ions Figure 3. Ranges and stopping power for fast and debris ions in cm graphite wall Fast Ions Debris Ions Figure 4. Ranges and stopping power for fast and debris ions in mm Li / cm CFC composite wall t t t t t < t < t < t t t t t t < t t < t t t t t t t < t t t t t t t < t t < t t t t t t < t < t < t The number of consistent ranges and the bound points are defined and used to calculate total deposition. Each ion is processed similarly. Knowing the arrival time of the ion i, one defines the range i i to which the ion belongs. After that, each pair t is considered from time mesh interval t j, j+ i i i t, t K tne. Suppose that both points belong to the standard mesh. If there are no time steps in between them, the deposition function at these points is increased by the values at these time points for ion i. If there are additional time steps between t i i,and t, then a simple linear interpolation is 4 j j+

29 i i performed to distribute all the energy deposited at time t in all points, including intermediate j, t j+ i ones. If some of the time points are not included in the standard mesh, say t, then again, a linear i i i interpolation of values at time t for some k is used, where belongs to the standard mesh. t j k, j+ Finally, separate ion deposition and total ion deposition functions are printed out in graphical forms. The former is as shown in Figure 5. t j k j CFC CX-U Figure 5. Ion energy deposition for CFC and tungsten Tungsten The fewer the number of points in a given ion spectra, the worse the computational accuracy of the ion deposition function. Figure 6 shows calculated results of the total deposition function for two cases involving a composite wall, consisting of mm lithium film on cm graphite. The first case uses a given number of points in the spectrum (from 5 to 5) for each incident ion. An example of such a spectrum for debris tritium ion is presented in Table 3. The second case demonstrates results of the same calculation, extrapolated to 9 points, preserving the total yield of each ion beam. Original histogram (5-5 points) Modified histogram (9 points) Figure 6. Ion energy deposition, for mm lithium / cm CFC 5

30 8. Laser Light Deposition Few studies have been conducted on laser light reflection and absorption, especially for high intensity beams. Some simple models have been developed for laser deposition into materials based on experimental results. The model discussed below, for laser light interaction with materials, is not only applicable for ICF reactor first-walls, but also for laser annealing of materials by laser pulses several nanoseconds long. This model will be coupled with the methods of solving the heat conduction equation with moving boundaries and calculating the dynamics of melting and evaporation. The deposition function for the laser radiation absorption in materials can be written as: q& ( x, = α P( x,, where α P ( x, The power absorbed can be written as: the absorption coefficient of the material; α ( t ) x ( R( ) F( e P( x, =, where the power absorbed at time t in the material per unit volume due to the laser pulse passing through it. F ( the incident power density; R ( the reflection coefficient. The absorption coefficient in some materials, like the reflectivity, strongly depends on the melting of the near-surface layer. As a result of previous work [34], the absorption coefficient as well as the reflection coefficient is made a function of time (whether the material is in solid or liquid phase) for a certain depth from the front surface. This depth depends on the absorption length of the material. Volumetric and surface depositions of laser energy depend on the wavelength of the laser. Depending on the target structure, a significant fraction of laser energy may be reflected from the target surface part of which is deposited in the first zone of the wall. After several reflections, most laser energy is deposited at the wall. Each deposition occurs at different times. Additionally, the volumetric and surface deposition power are calculated and used for the thermal evolution and temperature rise computation. 9. Thermal Evolution of the Chamber Wall The thermal response of a wall material exposed to thermonuclear radiation may be determined when all the time- and space-dependent energy-deposition functions are known. The aim of this section is to describe the numerical methods of solving the heat diffusion equation subject to several moving and boundary conditions. HEIGHTS-IFE can use several methods to solve this problem for double checking the thermal response: the Green's function approach, an approximate analytical solution for the nonlinear heat conduction equation using the perturbation theory in which thermal properties vary with temperature, and the finite difference/element approach for a very accurate solution with various flexible initial and boundary conditions. 9.. Integrated Deposition Model The general heat-transfer equation is given by T ρ c k T = q& ( x,, where t 6

31 ρ = ρ(t ) density of the material; c = c(t ) specific heat of the material; k = k(t ) thermal conductivity of the material. q& is the summation of all forms of energy deposited from photons, laser light, neutrons and ions. All the above properties vary with temperature. 9.. Thermal Response Models for Chamber First-walls The rapid heating of fusion first-wall components due to X-ray and ion debris deposition in ICF reactors may lead to melting and subsequently to intense evaporation. As a result, an accurate analysis of this heat conduction problem initially requires the solution of at least two moving boundary problems. A moving face where vaporization occurs becomes one boundary, in addition to the moving internal boundary between the liquid and solid. Because of the moving boundary and the difference between the properties of the liquid and solid states of the same material, the distribution is nonlinear Moving Boundary Care should be taken in solving moving boundary problems. These are difficult to solve and can present challenging mathematical and numerical questions. Whereas most moving boundary problems (also called Stefan problems) deal with melting, solidification, and slow evaporation where the interface is mathematically characterized by a fixed value of the temperature whose value is known in advance (such as melting and the boiling poin, problems involving intense evaporation or ablation must satisfy a moving boundary condition that is derived from energy and mass balances. As a result, these moving boundary conditions yield highly nonlinear equations whose determination is now an integral part of the solution for the entire problem Phase Change Consider the first-wall as a semi-infinite medium. This assumption is reasonable in view of the short heat penetration depth during a target debris deposition in ICF reactors. Under a deposition function q ( x, and an interface heat flux F (, the temperature distribution T s ( x, in the solid phase must then satisfy the heat conduction equation: ρ C s s T s t k T s s = q& ( x,, where ρ s C s k s density of the material; specific heat of the material; thermal conductivity of the material; q& ( x, volumetric energy deposition rate. All these thermophysical properties are functions of the local temperature. The boundary conditions are that T s (x, T b = constant at large depth distances x, and that on the surface x=, 7

32 4 4 ( Tv Ts F( = k s ( Tv ) + ρ s ( Tv ) Lvv( Tv ) + σ T x T v ( = T s (,; ), where L v latent heat; v ( T v ) velocity of the receding surface. The velocity of the receding surface is a function of the instantaneous surface temperature and other material parameters. The vaporization of the surface is assumed to be a continuous function of surface temperature. The radiative heat transfer term contains the Stefan-Boltzmann constant, σ, and the surface temperature, T, of the cold portion of the wall. This term is taken into account only for nonuniform deposition. However, in these symmetric calculations, the radiative heat transfer term is set to zero. Once melting occurs, the condensed phase consists of two regions: s( x m(, for melt layer, and m( x, for solid phase, where s( instantaneous location of the melted surface; m ( distance of the melted layer from the surface. The solid phase is treated as mentioned above, but the boundary condition at the solid-liquid interface x = m( is given by T ( x, = T ( x, = T, where T ( x, and T ( x, t are the temperatures s l m s l ) of the solid and liquid phases, respectively, and T m is the melting (or solidification) temperature. Then, Tl Ts d m( kl = k s + ρ s L f, where L f is the heat of fusion. x x d t m( m( The melting phase must satisfy the heat conduction equation, given by ρ C l l T l t k T l l = q& ( x, with the same boundary condition at the solid-liquid interface and the condition Tl F( = kl + ρl ( Tv ) L f v( at the surface x = s(. x s( Evaporation Moving Boundary If the heating of the wall is continued long enough and at a sufficiently high rate, significant vaporization may occur from the surface, assuming that the melting material stays in place. It is necessary to account for the receding surface at the interface between vapor and solid or liquid. This can be done by introducing a moving coordinate system, 8

33 z( = x s(, for which the surface always remains at z=. Transforming the heat conduction equation to this moving coordinate frame gives T T d z ρ C + k T t z d t dz dt ds( = = v(, dt = q& ( z,, where where v ( velocity of the receding surface. Substituting, one obtains T T ρ C ρcv( k T = q& ( z,. t z The main difference between this equation and that above is the presence of the convective term T v(. This term is important in the cases of intensive evaporation and vapor cloud formation if we z are to accurately calculate the temperature. The velocity of the receding surface v( is a highly nonlinear function of temperature. A complete solution of this problem is used in the computer code HEIGHTS-IFE Evaporation Models According to the Hertz-Knudsen-Langmuir theory of evaporation and condensation the net flux of atoms leaving the surface of the condensed phase is given by σ c Pe σ c Pc J = J e J c =, where πmkt J m k net vaporization flux; mass per atom; Boltzmann constant; σ e, σ c evaporation and condensation coefficients; P c P s ambient partial pressure in the chamber; saturation vapor pressure. 9

34 The coefficients σ e and σ c are used to compensate for nonideal evaporation and condensation. They are usually taken to be the same. The saturation vapor pressure is given by Ps is the derived constant, and H is the activation energy for evaporation. P H kt = P e, where If the surface temperature T of the condensed phase is different from the temperature T of the v ambient vapor, the evaporation and condensation fluxes may be written as J eq e c σ eps (Tv ) = πmkt σ c Pc eq J c =. The evaporation flux J e represents a maximum for evaporation into a vacuum, πmkt provided the vapor expands at a sufficient rate so the vapor density in front of the surface always remains low. This condition may not be valid in certain conditions. If the evaporation flux is expected to be high, the vapor density in front of the surface is finite, even if the vapor gas expands into vacuum. Meanwhile, the described approach may give preliminary estimations for both evaporation and condensation fluxes. More accurate transport calculations for intense evaporation have been performed by Anisimov and Rakhmatulina [35]. In their work the following problem was considered. The surface of a material, which occupies half-space is suddenly raised and held at a constant surface temperature T v for times t. The material begins to vaporize, and the vapor expands freely into the vacuum. Initially, the eq evaporation flux leaving the surface is equal to J e, but it decreases thereafter due to recondensation. This process of recondensation arises from two facts. First, the density of vapor expanding into a vacuum retains a finite value for t > in front of the surface. Second, atoms evaporated subsequently from the surface may collide with the already-present vapor phase and be backscattered toward the surface where they may be reabsorbed. The fraction of recondensing atoms increases as the vapor density and the spatial extension of the vapor phase increase with time. However, an asymptotic value of. is reached for this fraction after about collision times. The collision time τ c for the vapor atoms is given by τc =6 π na kt v m, where π a is the elastic scattering cross section for the vapor atoms and n is the vapor density in front of the surface. This equation is related to the maximum vacuum evaporation rate according to J eq e where the relation ktv = vn = n, 4 π m 8kTv v =, is used for the average velocity of the vapor atoms. π m For the elastic scattering cross section we may use the approximation atomic volume. Then, the collision time τ c is given by 4π a 3 3 v and = Ω, where Ω is the τ c 3 3 = 6 π Ω 4 3 J eq e. 3

35 Anisimov and Rakhmatulina have shown that the time-dependent evaporation rate may be approximated by t eq τ R J ( = J e.8 +.e, where τ R is the relaxation time for full condensation. The relaxation time τ R to reach, say, 98% of the full amount of recondensation after collision times τ c is given by τ = τ c or ln τ c τ R R 3 3 =.6 π Ω 4 eq In these equations, J e ( T v ) is a constant for t, since it was assumed that T v remains constant. For our application, however, the surface temperature T v ( varies with time. Nevertheless, the surface temperature rapidly approaches a saturation value once intense evaporation begins. Accordingly, the time variable t should be replaced by ( t, where the preheat time t may be estimated as follows. t v ) For instantaneous recondensation to become significant, the thickness of the vapor zone in front of the surface should be of the order of the mean free collision path l =. The thickness x of nπa material evaporated to produce a vapor zone of thickness l is then and in terms of, from the surface. a ( t ) 3 J eq e. v ( t ) x v = nl Ω x. 585 Ω 3 or ( t ) 4 x v = a. This corresponds roughly to a monolayer of atoms evaporated 3 v 3

36 Finally, the evaporation flux of atoms is equal to J J = J, eq e t tv ( eq τ c e ( Tv ( ).8 +.e and the velocity of the receding surface is v( = ΩJ (., for t t for t t v v 9.4. Condensation Model The condensation model is based on the surface conditions, target material properties, chamber design, and surface temperature. Condensation affects all processes, influences surface temperature evolution, and modifies surface properties. The net condensation rate can be calculated from the following equation: s ( P ( T ) αβ A P ) M G = sat sat c g, where π RT α, R M T s accommodation coefficient and universal gas constant; atomic mass; surface temperature of target material; P sat saturation pressure at temperature T s ; P g pressure of vapor in contact with the surface; β and scaling factors. A c. Variation of Thermal Properties with Temperature In the solution of the nonlinear heat conduction equation, all the thermophysical properties are allowed to vary with temperature. The advantage of HEIGHTS-IFE is that these material property variations are implemented as separate procedures with a known interface. There are no restrictions on the behavior of the properties, and the user may easily update, add, or modify these properties without affecting other parts of the HEIGHTS-IFE package. Material properties are divided into two sets. The first set of properties does not depend on temperature and known as constant parameters. The second set; saturated vapor pressure, density, specific heat, and thermal conductivity are the most important temperature-dependent material properties. Because of the diversity of methods to estimate and present this dependence, each material may have its own appropriate way of presentation. To increase the efficiency of the HEIGHTS-IFE package, all material-dependent parameters must be installed and checked at the very beginning of the solution for the heat conduction equation. Then, during the integration, the material properties are set. If a given property does not depend on temperature, the definition of material properties is done while the wall is described. Otherwise, the method of functional dependence and transmission is used. 3

37 Temperature-dependent material properties are implemented as an independent procedure, then transmitted to the main code as parameters. Each material property function is relevant to its particular material. The synopsis, however, must stay the same. As indicated in Table, each material implemented in HEIGHTS-IFE, has a unique code and short name. An example of the typical variation of material properties with temperature for a carbon-fiber composite is shown in Figure 7. The HEIGHTS-IFE material data base library contains up to different carbon based materials. Zone : CFC CX-U/Jap cm Atomic number: 6 Atomic mass: Melting Temperature: K Boiling Temperature: 395.5K Vapor. Enthalpy: 694. cal/mol or J/g Figure 7. Properties of CX-U carbon-fiber composite. Numerical Simulation of First-wall Temperature The chamber wall thermal evolution is also calculated in with HEIGHTS-IFE. The time evolution starts with the arrival of X-rays, then reflected laser light, then neutrons, then fast and slow ion debris. Also, in the case of gas-filled cavity, the reradiated absorbed gas energy can be taken into account as a surface heat flux. Table 5 lists all incident photon radiation and ion debris that contribute to the input energy flux and indicates how this energy is partitioned at the wall. The surface temperature is determined by both the boundary conditions and the kinetics of the evaporation process. The correct 33

38 boundary conditions entail partitioning of the incident energy flux into conduction, melting, evaporation, and possible radiation flux. The kinetics of evaporation establish the connection between the surface temperature and the net atom flux leaving the surface, taking into account the possibility of recondensation flux. Table 5. ARIES IFE reference target spectra at ns NRL Direct Drive Target (MJ) High-Yield Direct Drive Target (MJ) X-rays. (%) 6.7 (%) 5 (5%) Neutrons 9 (7%) 79 (69.75%) 36 (69%) Gammas.89 (.6%).7 (.4%).36 (.%) Burn Product Fast Ions 9.5 (3%) 5. (3%) 8.43 (%) Protons Deuterons Tritons He He Debris Ions kinetic energy 4.9 (6%) 6. (5%) 8. (4%) Protons Deuterons Tritons He He Be C Fe Br Gd Au Residual thermal energy Residual burn products Driver energy absorbed Total out 54 MJ 4 MJ 458 MJ HI Indirect Drive Target (MJ) 34

39 Carbon-fiber composite and tungsten are considered in this analysis because they are two of the main candidates for first-wall structural components in fusion reactors. Results of calculated surface temperature are presented in Figure 8. This calculation is for the bare-wall concept with no protection and for the lower yield NRL direct target spectra as represented in Table 5. Figure 8 shows the time evolution of the wall thermal response due to the sequence of different incident species. The 3-D distribution of the surface temperature in both time and depth is also shown in Figure 9.. Erosion Processes Figure 8. Surface temperature rise due to NRL direct drive target Under normal operation conditions, the main debris-surface interaction mechanisms are physical sputtering, chemical sputtering, and radiation-enhanced sublimation. High-Z materials, such as tungsten, show low effective sputtering yield at low plasma temperatures and, therefore, a preferable behavior. For higher ion temperatures, low-z materials such as lithium or carbon-fiber composites, the sputtering is less critical, but chemical erosion may become important and cause serious wall erosion... Physical Sputtering Physical sputtering involves the removal of surface atoms from a solid due to the impact of energetic particles. The sputtering process can be described by momentum transport in a collision cascade initiated by the incident particle in the surface layer of the solid. A surface atom is ejected if the cascade of atoms reaches the surface with energy larger than the surface binding energy [ 3]. Physical sputtering is measured by the sputtering yield Y, defined as the mean number of atoms removed from the surface layer of the wall per incident ion. The energy of incident ions must be larger than a threshold energy for sputtering to occur, which is determined by the surface binding energy E s E th of the atoms of the wall and the momentum transfer process. Sputtering yields and their dependence of the incident ion energy E, mass a, and angle has been investigated experimentally by many authors [3, 36, 37, 38]. 35

40 Figure 9. Temperature rise due to laser, X-ray, and ion depositions 36